Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are not limited to, the Hodge–Dirac operator, d+⋆d⋆{\displaystyle d+{\star }d{\star }} on a Riemannian manifold, the Dirac operator in euclidean space and its inverse on C0∞(Rn){\displaystyle C_{0}^{\infty }(\mathbf {R} ^{n})} and their conformal equivalents on the sphere, the Laplacian in euclidean n-space and the Atiyah–Singer–Dirac operator on a spin manifold, Rarita–Schwinger/Stein–Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on Spinc manifolds, systems of Dirac operators, the Paneitz operator, Dirac operators on hyperbolic space, the hyperbolic Laplacian and Weinstein equations.

Up to a sign this inverse is the Kelvin inverse of x. Solutions to the euclidean Dirac equation Df = 0 are called (left) monogenic functions. Monogenic functions are special cases of harmonic spinors on a spin manifold.

In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternionic analysis. When n = 4, the Dirac operator is sometimes referred to as the Cauchy–Riemann–Fueter operator. Further some aspects of Clifford analysis are referred to as hypercomplex analysis.

Suppose U′ is a domain in Rn−1 and g(x) is a Cℓn(C) valued real analytic function. Then g has a Cauchy–Kovalevskaia extension to the Dirac equation on some neighborhood of U′ in Rn. The extension is explicitly given by

Many Dirac type operators have a covariance under conformal change in metric. This is true for the Dirac operator in euclidean space, and the Dirac operator on the sphere under Möbius transformations. Consequently this holds true for Dirac operators on conformally flat manifolds and conformal manifolds which are simultaneously spin manifolds.

where △LB{\displaystyle \triangle _{LB}} is the Laplace–Beltrami operator on Sn. The operator △S{\displaystyle \triangle _{S}} is, via the Cayley transform, conformally equivalent to the euclidean Laplacian. Also

then we have a choice in sign in defining J(M, x). This means that for a conformally flat manifoldM we need a spin structure on M in order to define a spinor bundle on whose sections we can allow a Dirac operator to act. Explicit simple examples include the n-cylinder, the Hopf manifold obtained from n-euclidean space minus the origin, and generalizations of k-handled toruses obtained from upper half space by factoring it out by actions of generalized modular groups acting on upper half space totally discontinuously. A Dirac operator can be introduced in these contexts. These Dirac operators are special examples of Atiyah–Singer–Dirac operators.

Given a spin manifoldM with a spinor bundleS and a smooth section s(x) in S then, in terms of a local orthonormal basis e1(x), ..., en(x) of the tangent bundle of M, the Atiyah–Singer–Dirac operator acting on s is defined to be

where τ is the scalar curvature on the manifold, and Γ∗ is the adjoint of Γ. The operator D2 is known as the spinorial Laplacian.

If M is compact and τ ≥ 0 and τ > 0 somewhere then there are no non-trivial harmonic spinors on the manifold. This is Lichnerowicz' theorem. It is readily seen that Lichnerowicz' theorem is a generalization of Liouville's theorem from one variable complex analysis. This allows us to note that over the space of smooth spinor sections the operator D is invertible such a manifold.

In the cases where the Atiyah–Singer–Dirac operator is invertible on the space of smooth spinor sections with compact support one may introduce

Using Stokes' theorem, or otherwise, one can further determine that under a conformal change of metric the Dirac operators associated to each metric are proportional to each other, and consequently so are their inverses, if they exist.

All of this provides potential links to Atiyah–Singer index theory and other aspects of geometric analysis involving Dirac type operators.

In Clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or Poincaré metric.

For upper half space one splits the Clifford algebra, Cℓn into Cℓn−1+Cℓn-1en. So for a in Cℓn one may express a as b+cen with a, b in Cℓn−1. One then has projection operators P and Q defined as follows P(a) = b and Q(a) = c. The Hodge–Dirac operator acting on a function f with respect to the hyperbolic metric in upper half space is now defined to be

Rarita–Schwinger operators, also known as Stein–Weiss operators, arise in representation theory for the Spin and Pin groups. The operator Rk is a conformally covariant first order differential operator. Here k = 0, 1, 2, .... When k = 0, the Rarita–Schwinger operator is just the Dirac operator. In representation theory for the orthogonal group, O(n) it is common to consider functions taking values in spaces of homogeneous harmonic polynomials. When one refines this representation theory to the double covering Pin(n) of O(n) one replaces spaces of homogeneous harmonic polynomials by spaces of k homogeneous polynomial solutions to the Dirac equation, otherwise known as k monogenic polynomials. One considers a function f(x, u) where x in U, a domain in Rn, and u varies over Rn. Further f(x, u) is a k-monogenic polynomial in u. Now apply the Dirac operator Dx in x to f(x, u). Now as the Clifford algebra is not commutative Dxf(x, u) then this function is no longer k monogenic but is a homogeneous harmonic polynomial in u. Now for each harmonic polynomial hk homogeneous of degree k there is an Almansi–Fischer decomposition

hk(x)=pk(x)+xpk−1(x){\displaystyle h_{k}(x)=p_{k}(x)+xp_{k-1}(x)}

where pk and pk−1 are respectively k and k−1 monogenic polynomials. Let P be the projection of hk to pk then the Rarita–Schwinger operator is defined to be PDk, and it is denoted by Rk. Using Euler's Lemma one may determine that